Theorem eulerpartlemo 30427

Description: Lemma for eulerpart 30444: O is the set of odd partitions of N. (Contributed by Thierry Arnoux, 10-Aug-2017.)

Hypotheses

Ref	Expression
eulerpart.p	|- P = { f e. ( NN0 ^m NN ) | ( ( `' f " NN ) e. Fin /\ sum_ k e. NN ( ( f ` k ) x. k ) = N ) }
eulerpart.o	|- O = { g e. P | A. n e. ( `' g " NN ) -. 2 || n }
eulerpart.d	|- D = { g e. P | A. n e. NN ( g ` n ) <_ 1 }

Assertion

Ref	Expression
eulerpartlemo	|- ( A e. O <-> ( A e. P /\ A. n e. ( `' A " NN ) -. 2 || n ) )

Distinct variable groups:   g, n, A   P, g

Allowed substitution hints:   A( f, k)   D( f, g, k, n)   P( f, k, n)   N( f, g, k, n)   O( f, g, k, n)

Proof of Theorem eulerpartlemo

Step	Hyp	Ref	Expression
1		cnveq 5296	. . . 4 |- ( g = A -> `' g = `' A )
2	1	imaeq1d 5465	. . 3 |- ( g = A -> ( `' g " NN ) = ( `' A " NN ) )
3	2	raleqdv 3144	. 2 |- ( g = A -> ( A. n e. ( `' g " NN ) -. 2 || n <-> A. n e. ( `' A " NN ) -. 2 || n ) )
4		eulerpart.o	. 2 |- O = { g e. P | A. n e. ( `' g " NN ) -. 2 || n }
5	3, 4	elrab2 3366	1 |- ( A e. O <-> ( A e. P /\ A. n e. ( `' A " NN ) -. 2 || n ) )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   class class class wbr 4653   `'ccnv 5113   "cima 5117   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   1c1 9937   x. cmul 9941   <_ cle 10075   NNcn 11020   2c2 11070   NN0cn0 11292   sum_csu 14416   || cdvds 14983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  eulerpartlemr  30436